The incident occured at around 11.50 a m at the crossing of Bidhan Sarani and Kailash Bose Street, the police said.
A Kolkata Police officer said the bus suddenly picked up speed as the 35-year-old woman was alighting. This led to the woman losing her balance and she fell and slipped under the rear wheels of the bus, which crushed her right hand.
She was rushed to Calcutta Medical College and Hospital, where she is under treatment, he added.